# Recent Activity

## Shuffle-Exchange Conjecture ★★★

Author(s): Beneš; Folklore; Stone

Given integers , let be the smallest integer such that the symmetric group on the set of all words of length over a -letter alphabet can be generated as , where is the shuffle permutation defined by , and is the exchange group consisting of all permutations in preserving the first letters in the words.

Problem  (SE)   Find .
Conjecture  (SE)   .

Keywords:

## Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let be a positive integer. We say that a graph is strongly -colorable if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.

Conjecture   If is the maximal degree of a graph , then is strongly -colorable.

Keywords: strong coloring

## Friendly partitions ★★

Author(s): DeVos

A friendly partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

Problem   Is it true that for every , all but finitely many -regular graphs have friendly partitions?

Keywords: edge-cut; partition; regular

## Is there an algorithm to determine if a triangulated 4-manifold is combinatorially equivalent to the 4-sphere? ★★★

Author(s): Novikov

Problem   Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalent to the 4-sphere?

Keywords: 4-sphere; algorithm

## What is the homotopy type of the group of diffeomorphisms of the 4-sphere? ★★★★

Author(s): Smale

Problem   has the homotopy-type of a product space where is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of .

Keywords: 4-sphere; diffeomorphisms

## Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere? ★★★

Author(s): Kirby

Problem   Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.

Keywords: 3-manifold; 4-sphere; embedding

## Fundamental group torsion for subsets of Euclidean 3-space ★★

Author(s): Ancient/folklore

Problem   Does there exist a subset of such that its fundamental group has an element of finite order?

Keywords: subsets of euclidean space; torsion

## Which homology 3-spheres bound homology 4-balls? ★★★★

Author(s): Ancient/folklore

Problem   Is there a complete and computable set of invariants that can determine which (rational) homology -spheres bound (rational) homology -balls?

Keywords: cobordism; homology ball; homology sphere

## Realisation problem for the space of knots in the 3-sphere ★★

Author(s): Budney

Problem   Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .

Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .

There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .

Problem: What representations can be obtained?

Keywords: knot space; symmetry

## Slice-ribbon problem ★★★★

Author(s): Fox

Conjecture   Given a knot in which is slice, is it a ribbon knot?

Keywords: cobordism; knot; ribbon; slice

## Smooth 4-dimensional Poincare conjecture ★★★★

Author(s): Poincare; Smale; Stallings

Conjecture   If a -manifold has the homotopy type of the -sphere , is it diffeomorphic to ?

Keywords: 4-manifold; poincare; sphere

## Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

Problem   Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?

Keywords: 4-dimensional; Schoenflies; sphere

## Are different notions of the crossing number the same? ★★★

Author(s): Pach; Tóth

Problem   Does the following equality hold for every graph ?

The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the pairwise crossing number , we minimize the number of pairs of edges that cross.

Keywords: crossing number; pair-crossing number

## Shuffle-Exchange Conjecture (graph-theoretic form) ★★★

Author(s): Beneš; Folklore; Stone

Given integers , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).

Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).

Let be the smallest integer such that the graph is rearrangeable.

Problem   Find .
Conjecture   .

Keywords:

## Partition of Complete Geometric Graph into Plane Trees ★★

Author(s):

Conjecture   Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.

## Edge-Colouring Geometric Complete Graphs ★★

Question   What is the minimum number of colours such that every complete geometric graph on vertices has an edge colouring such that:
\item[Variant A] crossing edges get distinct colours, \item[Variant B] disjoint edges get distinct colours, \item[Variant C] non-disjoint edges get distinct colours, \item[Variant D] non-crossing edges get distinct colours.

Keywords: geometric complete graph, colouring

## Number of Cliques in Minor-Closed Classes ★★

Author(s): Wood

Question   Is there a constant such that every -vertex -minor-free graph has at most cliques?

Keywords: clique; graph; minor

## A gold-grabbing game ★★

Author(s): Rosenfeld

Setup Fix a tree and for every vertex a non-negative integer which we think of as the amount of gold at .

2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex of the tree, takes the gold at this vertex, and then deletes . The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.

Problem   Find optimal strategies for the players.

Keywords: game; tree

## Circular colouring the orthogonality graph ★★

Author(s): DeVos; Ghebleh; Goddyn; Mohar; Naserasr

Let denote the graph with vertex set consisting of all lines through the origin in and two vertices adjacent in if they are perpendicular.

Problem   Is ?

## Crossing numbers and coloring ★★★

Author(s): Albertson

We let denote the crossing number of a graph .

Conjecture   Every graph with satisfies .

Keywords: coloring; complete graph; crossing number